Using FaceTime http://support.apple.com/kb/ht4319
Troubleshooting FaceTime http://support.apple.com/kb/TS3367
The Complete Guide to FaceTime + iMessage: Setup, Use, and Troubleshooting
Troubleshooting FaceTime and iMessage activation
Using FaceTime and iMessage behind a firewall
iOS: About Messages
Set up iMessage
iOS 6 and OS X Mountain Lion: Link your phone number and Apple ID for use with FaceTime and iMessage
How to Set Up & Use iMessage on iPhone, iPad, & iPod touch with iOS
Troubleshooting iMessage Issues: Some Useful Tips You Should Try
Setting Up Multiple iOS Devices for iMessage and Facetime
FaceTime and iMessage not accepting Apple ID password
Unable to use FaceTime and iMessage with my apple ID
How to Block Someone on FaceTime
My Facetime Doesn't Ring
To send messages to non-Apple devices, check out the TextFree app https://itunes.apple.com/us/app/text-free-textfree-sms-real/id399355755?mt=8
How to Send SMS from iPad
You can check the status of the FaceTime/iMessage servers at this link.
Try the usual steps: restart, reset, restore.
If you try restoring from a backup and that doesn't fix the problem, try restoring to factory settings and synching your apps. You'll lose the app data and settings, but if the problem is due to a corrupt cache or settings file, that should cure it.